HEAT TRANSFER A Practical Approach

Transcription

1 Complete Solution Manual to Accompany HEAT TRANSFER A Practical Approach SECOND EDITION YUNUS A. CENGEL

2 Preface This manual is prepared as an aide to the instructors in correcting homework assignments, but it can also be used as a source of additional example problems for use in the classroom. With this in mind, all solutions are prepared in full detail in a systematic manner, using a word processor with an equation editor. The solutions are structured into the following sections to make it easy to locate information and to follow the solution procedure, as appropriate: Solution - The problem is posed, and the quantities to be found are stated. Assumptions - The significant assumptions in solving the problem are stated. Properties - The material properties needed to solve the problem are listed. Analysis - The problem is solved in a systematic manner, showing all steps. Discussion - Comments are made on the results, as appropriate. A sketch is included with most solutions to help the students visualize the physical problem, and also to enable the instructor to glance through several types of problems quickly, and to make selections easily. Problems designated with the CD icon in the text are also solved with the EES software, and electronic solutions complete with parametric studies are available on the CD that accompanies the text. Comprehensive problems designated with the computer-ees icon [pick one of the four given] are solved using the EES software, and their solutions are placed at the Instructor Manual section of the Online Learning Center (OLC at Access to solutions is limited to instructors only who adopted the text, and instructors may obtain their passwords for the OLC by contacting their McGraw-Hill Sales Representative at Every effort is made to produce an error-free Solutions Manual. However, in a text of this magnitude, it is inevitable to have some, and we will appreciate hearing about them. We hope the text and this Manual serve their purpose in aiding with the instruction of Heat Transfer, and making the Heat Transfer experience of both the instructors and students a pleasant and fruitful one. We acknowledge, with appreciation, the contributions of numerous users of the first edition of the book who took the time to report the errors that they discovered. All of their suggestions have been incorporated. Special thanks are due to Dr. Mehmet Kanoglu who checked the accuracy of most solutions in this Manual. Yunus A. Çengel July 00

3 Thermodynamics and Heat Transfer Chapter 1 BASICS OF HEAT TRANSFER Chapter 1 Basics of Heat Transfer 1-1C Thermodynamics deals with the amount of heat transfer as a system undergoes a process from one equilibrium state to another. Heat transfer, on the other hand, deals with the rate of heat transfer as well as the temperature distribution within the system at a specified time. 1-C (a The driving force for heat transfer is the temperature difference. (b The driving force for electric current flow is the electric potential difference (voltage. (a The driving force for fluid flow is the pressure difference. 1-C The caloric theory is based on the assumption that heat is a fluid-like substance called the "caloric" which is a massless, colorless, odorless substance. It was abandoned in the middle of the nineteenth century after it was shown that there is no such thing as the caloric. 1-C The rating problems deal with the determination of the heat transfer rate for an existing system at a specified temperature difference. The sizing problems deal with the determination of the size of a system in order to transfer heat at a specified rate for a specified temperature difference. 1-5C The experimental approach (testing and taking measurements has the advantage of dealing with the actual physical system, and getting a physical value within the limits of experimental error. However, this approach is expensive, time consuming, and often impractical. The analytical approach (analysis or calculations has the advantage that it is fast and inexpensive, but the results obtained are subject to the accuracy of the assumptions and idealizations made in the analysis. 1-6C Modeling makes it possible to predict the course of an event before it actually occurs, or to study various aspects of an event mathematically without actually running expensive and time-consuming experiments. When preparing a mathematical model, all the variables that affect the phenomena are identified, reasonable assumptions and approximations are made, and the interdependence of these variables are studied. The relevant physical laws and principles are invoked, and the problem is formulated mathematically. Finally, the problem is solved using an appropriate approach, and the results are interpreted. 1-7C The right choice between a crude and complex model is usually the simplest model which yields adequate results. Preparing very accurate but complex models is not necessarily a better choice since such models are not much use to an analyst if they are very difficult and time consuming to solve. At the minimum, the model should reflect the essential features of the physical problem it represents. 1-1

4 Heat and Other Forms of Energy 1-8C The rate of heat transfer per unit surface area is called heat flux &q. It is related to the rate of heat transfer by Q & qda &. A 1-9C Energy can be transferred by heat, work, and mass. An energy transfer is heat transfer when its driving force is temperature difference. 1-10C Thermal energy is the sensible and latent forms of internal energy, and it is referred to as heat in daily life. 1-11C For the constant pressure case. This is because the heat transfer to an ideal gas is mc p ΔT at constant pressure and mc p ΔT at constant volume, and C p is always greater than C v. 1-1 A cylindrical resistor on a circuit board dissipates 0.6 W of power. The amount of heat dissipated in h, the heat flux, and the fraction of heat dissipated from the top and bottom surfaces are to be determined. Assumptions Heat is transferred uniformly from all surfaces. Analysis (a The amount of heat this resistor dissipates during a -hour period is Q Q & Δt (0.6 W( h 1. Wh 51.8 kj (since 1 Wh 600 Ws.6 kj Q & (b The heat flux on the surface of the resistor is πd π (0. cm A s + πdl + π (0. cm(1.5 cm cm 0.60 W q& s W/cm A.16 cm s (c Assuming the heat transfer coefficient to be uniform, heat transfer is proportional to the surface area. Then the fraction of heat dissipated from the top and bottom surfaces of the resistor becomes Qtop base Atop base or (11.8% Qtotal Atotal 16. Discussion Heat transfer from the top and bottom surfaces is small relative to that transferred from the side surface. Resistor 0.6 W 1-

5 1-1E A logic chip in a computer dissipates W of power. The amount heat dissipated in 8 h and the heat flux on the surface of the chip are to be determined. Assumptions Heat transfer from the surface is uniform. Analysis (a The amount of heat the chip dissipates during an 8-hour period is Q Q & Δt ( W ( 8 h Wh 0.0 kwh Logic chip Q & W (b The heat flux on the surface of the chip is q& s A s W 0.08 in 7.5 W/in 1-1 The filament of a 150 W incandescent lamp is 5 cm long and has a diameter of 0.5 mm. The heat flux on the surface of the filament, the heat flux on the surface of the glass bulb, and the annual electricity cost of the bulb are to be determined. Assumptions Heat transfer from the surface of the filament and the bulb of the lamp is uniform. Analysis (a The heat transfer surface area and the heat flux on the surface of the filament are A s πdl π ( 0.05 cm(5 cm cm Q & 150 W 191 W/cm W/m Lamp q& s A 150 W s cm (b The heat flux on the surface of glass bulb is A s πd π ( 8 cm 01.1cm 150 W q& s 0.75 W/cm 7500 W/m As 01.1cm (c The amount and cost of electrical energy consumed during a one-year period is Electricity Consumption Δt (. 015 kw(65 8 h / yr 8 kwh / yr Annual Cost (8 kwh / yr($0.08 / kwh $5.0 / yr 1-15 A 100 W iron is left on the ironing board with its base exposed to the air. The amount of heat the iron dissipates in h, the heat flux on the surface of the iron base, and the cost of the electricity are to be determined. Assumptions Heat transfer from the surface is uniform. Analysis (a The amount of heat the iron dissipates during a -h period is Q Q & Δt (. 1 kw( h. kwh (b The heat flux on the surface of the iron base is Q & (.( base W 1080 W base 1080 W q& 7,000 W / m Abase m (c The cost of electricity consumed during this period is Cost of electricity (. kwh ($0.07 / kwh $0.17 Iron 100 W 1-

6 1-16 A 15 cm 0 cm circuit board houses 10 closely spaced 0.1 W logic chips. The amount of heat dissipated in 10 h and the heat flux on the surface of the circuit board are to be determined. Assumptions 1 Heat transfer from the back surface of the board is negligible. Heat transfer from the front surface is uniform. Analysis (a The amount of heat this circuit board dissipates during a 10-h period is Q & ( 10(0.1 W 1. W Q Q & Δt (0.01 kw(10 h 0.1kWh (b The heat flux on the surface of the circuit board is A s q& s ( 0.15 m(0. m 0.0m A s 1. W 0.0 m 80 W/m 0 cm 1-17 An aluminum ball is to be heated from 80 C to 00 C. The amount of heat that needs to be transferred to the aluminum ball is to be determined. Assumptions The properties of the aluminum ball are constant. Properties The average density and specific heat of aluminum are given to be ρ,700 kg/m and C kj/kg. C. p 15 cm Analysis The amount of energy added to the ball is simply the change in its internal energy, and is determined from Etransfer Δ U mc( T T1 where π π m ρv ρ D ( 700 kg / m ( 015. m E. 77 kg 6 6 Substituting, E transfer ( 77. kg(0.90 kj / kg. C(00-80 C 515 kj Therefore, 515 kj of energy (heat or work such as electrical energy needs to be transferred to the aluminum ball to heat it to 00 C. Chips, 0.1 W Metal ball Q & 1-18 The body temperature of a man rises from 7 C to 9 C during strenuous exercise. The resulting increase in the thermal energy content of the body is to be determined. Assumptions The body temperature changes uniformly. Properties The average specific heat of the human body is given to be.6 kj/kg. C. Analysis The change in the sensible internal energy content of the body as a result of the body temperature rising C during strenuous exercise is ΔU mcδt (70 kg(.6 kj/kg. C( C 50 kj 1-

7 1-19 An electrically heated house maintained at C experiences infiltration losses at a rate of 0.7 ACH. The amount of energy loss from the house due to infiltration per day and its cost are to be determined. Assumptions 1 Air as an ideal gas with a constant specific heats at room temperature. The volume occupied by the furniture and other belongings is negligible. The house is maintained at a constant temperature and pressure at all times. The infiltrating air exfiltrates at the indoors temperature of C. Properties The specific heat of air at room temperature is C p kj/kg. C (Table A-15. Analysis The volume of the air in the house is V ( floor space(height ( 00 m ( m 600 m Noting that the infiltration rate is 0.7 ACH (air changes per hour and thus the air in the house is completely replaced by the outdoor air times per day, the mass flow rate of air through the house due to infiltration is PoV& air Po (ACH Vhouse m& air RT RT o o (89.6 kpa( m / day 11,1 kg/day (0.87 kpa.m /kg.k( K 0.7 ACH Noting that outdoor air enters at 5 C and leaves at C, the energy loss of this house per day is Q & infilt m& aircp ( Tindoors Toutdoors (11,1 kg/day(1.007 kj/kg. C( 5 C 19,681 kj/day 5.8 kwh/day At a unit cost of $0.08/kWh, the cost of this electrical energy lost by infiltration is Enegy Cost (Energy used(unit cost of energy (5.8 kwh/day($0.08/kwh $.1/day 5 C C AIR 1-5

8 1-0 A house is heated from 10 C to C by an electric heater, and some air escapes through the cracks as the heated air in the house expands at constant pressure. The amount of heat transfer to the air and its cost are to be determined. Assumptions 1 Air as an ideal gas with a constant specific heats at room temperature. The volume occupied by the furniture and other belongings is negligible. The pressure in the house remains constant at all times. Heat loss from the house to the outdoors is negligible during heating. 5 The air leaks out at C. Properties The specific heat of air at room temperature is C p kj/kg. C (Table A-15. Analysis The volume and mass of the air in the house are V ( floor space(height ( 00 m ( m 600 m m PV ( 101. kpa(600 m kg RT ( kpa.m / kg.k( K Noting that the pressure in the house remains constant during heating, the amount of heat that must be transferred to the air in the house as it is heated from 10 to C is determined to be Q mc ( T T1 (77.9 kg(1.007 kj/kg. C( 10 C 908 kj p Noting that 1 kwh 600 kj, the cost of this electrical energy at a unit cost of $0.075/kWh is Enegy Cost (Energy used(unit cost of energy (908 / 600 kwh($0.075/kwh 10 C AIR $0.19 Therefore, it will cost the homeowner about 19 cents to raise the temperature in his house from 10 to C. C 1-1E A water heater is initially filled with water at 5 F. The amount of energy that needs to be transferred to the water to raise its temperature to 10 F is to be determined. Assumptions 1 Water is an incompressible substance with constant specific heats at room temperature. No water flows in or out of the tank during heating. Properties The density and specific heat of water are given to be 6 lbm/ft and 1.0 Btu/lbm. F. Analysis The mass of water in the tank is 1ft m ρv (6 lbm/ft (60 gal 97. lbm 7.8 gal Then, the amount of heat that must be transferred to the water in the tank as it is heated from 5 to10 F is determined to be Q mc( T 1 T (97. lbm(1.0 Btu/lbm. F(10 5 F 7,50 Btu 10 F 5 F Water The First Law of Thermodynamics 1-C Warmer. Because energy is added to the room air in the form of electrical work. 1-C Warmer. If we take the room that contains the refrigerator as our system, we will see that electrical work is supplied to this room to run the refrigerator, which is eventually dissipated to the room as waste heat. 1-6

9 1-C Mass flow rate &m is the amount of mass flowing through a cross-section per unit time whereas the volume flow rate V & is the amount of volume flowing through a cross-section per unit time. They are related to each other by m& ρ V& where ρ is density. 1-5 Two identical cars have a head-on collusion on a road, and come to a complete rest after the crash. The average temperature rise of the remains of the cars immediately after the crash is to be determined. Assumptions 1 No heat is transferred from the cars. All the kinetic energy of cars is converted to thermal energy. Properties The average specific heat of the cars is given to be 0.5 kj/kg. C. Analysis We take both cars as the system. This is a closed system since it involves a fixed amount of mass (no mass transfer. Under the stated assumptions, the energy balance on the system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 system Change in internal, kinetic, potential, etc. energies 0 ΔUcars + ΔKEcars 0 ( mcδt cars + [ m( 0 V / ] cars That is, the decrease in the kinetic energy of the cars must be equal to the increase in their internal energy. Solving for the velocity and substituting the given quantities, the temperature rise of the cars becomes mv ΔT mc / V C / (90,000 / 600 m/s 0.5 kj/kg. C / 1kJ/kg 1000 m /s 0.69 C 1-6 A classroom is to be air-conditioned using window air-conditioning units. The cooling load is due to people, lights, and heat transfer through the walls and the windows. The number of 5-kW window air conditioning units required is to be determined. Assumptions There are no heat dissipating equipment (such as computers, TVs, or ranges in the room. Analysis The total cooling load of the room is determined from + + where lights people heatgain cooling lights people heat gain W 1 kw 0 60kJ/h 1,00 kj/h kw 15,000 kj/h.17 kw Substituting, & kw Q cooling Thus the number of air-conditioning units required is 9.17 kw 1.8 units 5 kw/unit 15,000 kj/h Room 0 people 10 bulbs Q cool 1-7

10 1-7E The air in a rigid tank is heated until its pressure doubles. The volume of the tank and the amount of heat transfer are to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δpe Δke 0. Constant specific heats at room temperature can be used for air. This assumption results in negligible error in heating and air-conditioning applications. Properties The gas constant of air is R 0.70 psia.ft /lbm.r Btu/lbm.R (Table A-1. Analysis (a We take the air in the tank as our system. This is a closed system since no mass enters or leaves. The volume of the tank can be determined from the ideal gas relation, mrt V P 1 1 (0lbm(0.70psia ft /lbm R( R 80.0ft 50psia (b Under the stated assumptions and observations, the energy balance becomes Ein Eout ΔE 1 system 1 Net energy transfer by heat, work, and mass Change in internal, kinetic, potential, etc. energies Q Δ U Q m( u u mc ( T T in in 1 v 1 The final temperature of air is PV 1 PV P T T T P T 1 (50 R 1080 R 1 1 The specific heat of air at the average temperature of T ave ( / 810 R 50 F is C v,ave C p,ave R Btu/lbm.R. Substituting, Q (0 lbm( Btu/lbm.R( R 1890 Btu Air 0 lbm 50 psia 80 F Q 1-8

11 1-8 The hydrogen gas in a rigid tank is cooled until its temperature drops to 00 K. The final pressure in the tank and the amount of heat transfer are to be determined. Assumptions 1 Hydrogen is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -0 C and 1.0 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Properties The gas constant of hydrogen is R.1 kpa.m /kg.k (Table A-1. Analysis (a We take the hydrogen in the tank as our system. This is a closed system since no mass enters or leaves. The final pressure of hydrogen can be determined from the ideal gas relation, P1 V PV T 00 K P P1 (50 kpa kpa T1 T T1 0 K (b The energy balance for this system can be expressed as Ein Eout ΔE 1 system 1 where Net energy transfer by heat, work, and mass 1 1 Qout ΔU Q Δ U m( u u mc ( T T out Change in internal, kinetic, potential, etc. energies 1 v 1 P V (50 kpa(1.0 m m 0.1 kg RT (.1 kpa m /kg K(0 K H 50 kpa 0 K Using the C v (C p R kj/kg.k value at the average temperature of 60 K and substituting, the heat transfer is determined to be Q out (0.1 kg(10.9 kj/kg K(0-00K kj Q 1-9

12 1-9 A resistance heater is to raise the air temperature in the room from 7 to 5 C within 0 min. The required power rating of the resistance heater is to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air. This assumption results in negligible error in heating and air-conditioning applications. Heat losses from the room are negligible. Properties The gas constant of air is R 0.87 kpa.m /kg.k (Table A-1. Also, C p kj/kg K for air at room temperature (Table A-15. Analysis We observe that the pressure in the room remains constant during this process. Therefore, some air will leak out as the air expands. However, we can take the air to be a closed system by considering the air in the room to have undergone a constant pressure expansion process. The energy balance for this steady-flow system can be expressed as E E ΔE or, 1 in out Net energy transfer by heat, work, and mass 1 Wein, Wb ΔU W, ΔH m( h h1 mc ( T T1 ein system Change in internal, kinetic, potential, etc. energies W& Δ t mc ( T T ein, p, ave 1 The mass of air is V m m PV 1 (100 kpa(10 m 19. kg RT1 (0.87 kpa m / kg K(80 K Using C p value at room temperature, the power rating of the heater becomes o o W (19. kg(1.007 kj/kg C(5 7 C/(15 60 s.01 kw & e, in p W e 5 6 m 7 C AIR 1-10

13 1-0 A room is heated by the radiator, and the warm air is distributed by a fan. Heat is lost from the room. The time it takes for the air temperature to rise to 0 C is to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air, C p and C v 0.70 kj/kg K. This assumption results in negligible error in heating and air-conditioning applications. The local atmospheric pressure is 100 kpa. Properties The gas constant of air is R 0.87 kpa.m /kg.k (Table A-1. Also, C p kj/kg K for air at room temperature (Table A-15. Analysis We take the air in the room as the system. This is a closed system since no mass crosses the system boundary during the process. We observe that the pressure in the room remains constant during this process. Therefore, some air will leak out as the air expands. However we can take the air to be a closed system by considering the air in the room to have undergone a constant pressure process. The energy balance for this system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 system Change in internal, kinetic, potential, etc. energies Qin + We, in Wb Qout ΔU 5,000 kj/h ( Q & in + W & e in Q &, out Δt ΔH m( h h1 mcp ( T T1 ROOM The mass of air is V m m PV m 5m 7m 1 (100 kpa(10 m 17. kg RT1 (0.87 kpa m / kg K(8 K Steam Using the C p value at room temperature, 10,000 kj/h W [( 10, /600 kj/s kj/s] pw Δt (17. kg(1.007 kj/kg C(0 10 C It yields Δt 116 s 1-11

14 1-1 A student living in a room turns his 150-W fan on in the morning. The temperature in the room when she comes back 10 h later is to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air. This assumption results in negligible error in heating and air-conditioning applications. All the doors and windows are tightly closed, and heat transfer through the walls and the windows is disregarded. Properties The gas constant of air is R 0.87 kpa.m /kg.k (Table A-1. Also, C p kj/kg K for air at room temperature (Table A-15 and C v C p R 0.70 kj/kg K. Analysis We take the room as the system. This is a closed system since the doors and the windows are said to be tightly closed, and thus no mass crosses the system boundary during the process. The energy balance for this system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 system Change in internal, kinetic, potential, etc. energies Wein, ΔU Wein, m( u u1 mcv( T T1 The mass of air is V m m PV 1 (100 kpa(1 m 17. kg RT1 (0.87 kpa m / kg K(88 K The electrical work done by the fan is We W& e Δ t (0.15 kj / s( s 500 kj Substituting and using C v value at room temperature, 500 kj (17. kg(0.70 kj/kg. C(T - 15 C T 58.1 C W e (insulated ROOM m 6 m 6 m 1-1

15 1-E A paddle wheel in an oxygen tank is rotated until the pressure inside rises to 0 psia while some heat is lost to the surroundings. The paddle wheel work done is to be determined. Assumptions 1 Oxygen is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -181 F and 76 psia. The kinetic and potential energy changes are negligible, Δke Δpe 0. The energy stored in the paddle wheel is negligible. This is a rigid tank and thus its volume remains constant. Properties The gas constant of oxygen is R 0.5 psia.ft /lbm.r Btu/lbm.R (Table A-1E. Analysis We take the oxygen in the tank as our system. This is a closed system since no mass enters or leaves. The energy balance for this system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass W Q ΔU pw, in 1 out system Change in internal, kinetic, potential, etc. energies W, Q + m( u u Q + mc ( T T pw in out 1 out v 1 The final temperature and the number of moles of oxygen are PV 1 PV P T T T P T 0 psia 1 (50 R 75 R psia m PV 1 (1.7 psia(10 ft 0.81 lbm RT (0.5 psia ft / lbmol R(50 R 1 O 1.7 psia 80 F 0 Btu The specific heat ofoxygen at the average temperature of T ave (75+50/ 68 R 178 F is C v,ave C p R Btu/lbm.R. Substituting, W pw,in (0 Btu + (0.81 lbm(0160 Btu/lbm.R(75-50 Btu/lbmol 5. Btu Discussion Note that a cooling fan actually causes the internal temperature of a confined space to rise. In fact, a 100-W fan supplies a room as much energy as a 100-W resistance heater. 1- It is observed that the air temperature in a room heated by electric baseboard heaters remains constant even though the heater operates continuously when the heat losses from the room amount to 7000 kj/h. The power rating of the heater is to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. We the temperature of the room remains constant during this process. Analysis We take the room as the system. The energy balance in this case reduces to E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 We, in Qout ΔU 0 W Q e, in out system Change in internal, kinetic, potential, etc. energies since ΔU mc v ΔT 0 for isothermal processes of ideal gases. Thus, W & e, in out 1kW 7000kJ/h 1.9 kw 600kJ/h AIR W e 1-1

16 1- A hot copper block is dropped into water in an insulated tank. The final equilibrium temperature of the tank is to be determined. Assumptions 1 Both the water and the copper block are incompressible substances with constant specific heats at room temperature. The system is stationary and thus the kinetic and potential energy changes are zero, ΔKE ΔPE 0 and ΔE ΔU. The system is well-insulated and thus there is no heat transfer. Properties The specific heats of water and the copper block at room temperature are C p, water.18 kj/kg C and C p, Cu 0.86 kj/kg C (Tables A- and A-9. Analysis We observe that the volume of a rigid tank is constant We take the entire contents of the tank, water + copper block, as the system. This is a closed system since no mass crosses the system boundary during the process. The energy balance on the system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 0 ΔU or, ΔU + ΔU 0 Cu system Change in internal, kinetic, potential, etc. energies water [ mc ( T T ] + [ mc( T T ] 0 1 Cu 1 water Using specific heat values for copper and liquid water at room temperature and substituting, (50 kg(0.86 kj/kg C( T 70 C + (80 kg(.18 kj/kg C( T 5 C 0 T 7.5 C WATER Copper 1-5 An iron block at 100 C is brought into contact with an aluminum block at 00 C in an insulated enclosure. The final equilibrium temperature of the combined system is to be determined. Assumptions 1 Both the iron and aluminum block are incompressible substances with constant specific heats. The system is stationary and thus the kinetic and potential energy changes are zero, ΔKE ΔPE 0 and ΔE ΔU. The system is well-insulated and thus there is no heat transfer. Properties The specific heat of iron is given in Table A- to be 0.5 kj/kg. C, which is the value at room temperature. The specific heat of aluminum at 50 K (which is somewhat below 00 C 7 K is 0.97 kj/kg. C. Analysis We take the entire contents of the enclosure iron + aluminum blocks, as the system. This is a closed system since no mass crosses the system boundary during the process. The energy balance on the system can be expressed as Ein Eout ΔE 1 1 Net energy transfer by heat, work, and mass ΔU iron 0 ΔU system Change in internal, kinetic, potential, etc. energies + ΔU 0 or, [ mc ( T T ] + [ mc( T T ] 0 Al 1 iron 1 Al Substituting, (0 kg(0.50 kj / kg oc( T 100 oc + (0 kg(0.97 kj / kg oc( T 00 oc 0 T 168 C 1-6 An unknown mass of iron is dropped into water in an insulated tank while being stirred by a 00-W paddle wheel. Thermal equilibrium is established after 5 min. The mass of the iron is to be determined. Assumptions 1 Both the water and the iron block are incompressible substances with constant specific heats at room temperature. The system is stationary and thus the kinetic and potential energy changes are zero, ΔKE ΔPE 0 and ΔE ΔU. The system is well-insulated and thus there is no heat transfer. 0 kg Al 0 kg iron 1-1

17 Properties The specific heats of water and the iron block at room temperature are C p, water.18 kj/kg C and C p, iron 0.5 kj/kg C (Tables A- and A-9. The density of water is given to be 1000 kg/m³. Analysis We take the entire contents of the tank, water + iron block, as the system. This is a closed system since no mass crosses the system boundary during the process. The energy balance on the system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 W pw,in ΔU system Change in internal, kinetic, potential, etc. energies or, W ΔU + Δ U where W pw, in pw,in iron water [ mc( T T 1 ] iron + [ mc( T T 1 ] water mwater ρv (1000 kg / m (0.08 m 80 kg W W& Δt (0. kj / s(5 60 s 00 kj pw pw Using specific heat values for iron and liquid water and substituting, (00kJ m iron (0.5 kj/kg C(7 90 C + (80 kg(.18 kj/kg C(7 0 C 0 m iron 7.1 kg W pw WATER Iron 1-15

18 1-7E A copper block and an iron block are dropped into a tank of water. Some heat is lost from the tank to the surroundings during the process. The final equilibrium temperature in the tank is to be determined. Assumptions 1 The water, iron, and copper blocks are incompressible substances with constant specific heats at room temperature. The system is stationary and thus the kinetic and potential energy changes are zero, ΔKE ΔPE 0 and ΔE ΔU. Properties The specific heats of water, copper, and the iron at room temperature are C p, water 1.0 Btu/lbm F, C p, Copper 0.09 Btu/lbm F, and C p, iron Btu/lbm F (Tables A-E and A-9E. Analysis We take the entire contents of the tank, water + iron + copper blocks, as the system. This is a closed system since no mass crosses the system boundary during the process. The energy balance on the system can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass 1 system Change in internal, kinetic, potential, etc. energies Q ΔU ΔU + ΔU + ΔU out copper iron water or Q out [ mc( T T 1 ] copper + [ mc( T T 1 ] iron + [ mc( T T 1 ] water WATER Using specific heat values at room temperature for simplicity and Copper substituting, 600Btu (90lbm(0.09Btu/lbm F( T 160 F + (50lbm(0.107Btu/lbm F( T 00 F + (180lbm(1.0Btu/lbm F( T 70 F T 7. F Iron 600 kj 1-16

19 1-8 A room is heated by an electrical resistance heater placed in a short duct in the room in 15 min while the room is losing heat to the outside, and a 00-W fan circulates the air steadily through the heater duct. The power rating of the electric heater and the temperature rise of air in the duct are to be determined.. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air. This assumption results in negligible error in heating and air-conditioning applications. Heat loss from the duct is negligible. The house is air-tight and thus no air is leaking in or out of the room. Properties The gas constant of air is R 0.87 kpa.m /kg.k (Table A-1. Also, C p kj/kg K for air at room temperature (Table A-15 and C v C p R 0.70 kj/kg K. Analysis (a We first take the air in the room as the system. This is a constant volume closed system since no mass crosses the system boundary. The energy balance for the room can be expressed as E E ΔE 1 in out Net energy transfer by heat, work, and mass W + W Q ΔU e,in fan,in out system Change in internal, kinetic, potential, etc. energies ( W & + W & Q & Δt m( u u mc ( T T e,in fan,in out The total mass of air in the room is 1 1 v m 00 kj/min V 5 6 8m 0m P1 V ( 98kPa( 0m W m 8.6kg RT 1 ( 0.87kPa m /kg K( 88K 00 W Then the power rating of the electric heater is determined to be W& e, in out W& fan,in + mcv ( T T1 / Δt ( 00/60kJ/s ( 0.kJ/s + ( 8.6kg (0.70 kj/kg C ( 5 15 C/ ( 15 60s 5.1 kw (b The temperature rise that the air experiences each time it passes through the heater is determined by applying the energy balance to the duct, E& & in Eout W& W& mh & 0 e,in + fan,in + 1 out + mh & (since Δke Δpe 0 W& + W& e,in fan,in m& Δh mc & pδt Thus, W& e,in + W& fan,in ( kj/s ΔT 6.7 C mc & p ( 50/60kg/s( kj/kg K 1-17

20 1-9 The resistance heating element of an electrically heated house is placed in a duct. The air is moved by a fan, and heat is lost through the walls of the duct. The power rating of the electric resistance heater is to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air. This assumption results in negligible error in heating and air-conditioning applications. Properties The specific heat of air at room temperature is C p kj/kg C (Table A-15. Analysis We take the heating duct as the system. This is a control volume since mass crosses the system boundary during the process. We observe that this is a steady-flow process since there is no change with time at any point and thus ΔmCV 0 and ΔECV 0. Also, there is only one inlet and one exit and thus m& m& m&. The energy balance for this steady-flow system can be expressed in the rate form as 1 & & & 0 (steady E E ΔE E& E& 1 system 0 1 in out in out Rate of net energy transfer by heat, work, and mass Rate of change in internal, kinetic, potential, etc. energies W& + W& + mh & + mh & (since Δke Δpe 0 e,in fan,in 1 out W& W& + mc & ( T T e,in out fan,in p 1 Substituting, the power rating of the heating element is determined to be W & e,in (0.5 kw (0. kw + ( 0.6 kg/s( 1.007kJ/kg C( 5 C.97 kw 50 W W 00 W 1-18

21 1-0 Air is moved through the resistance heaters in a 100-W hair dryer by a fan. The volume flow rate of air at the inlet and the velocity of the air at the exit are to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air. The power consumed by the fan and the heat losses through the walls of the hair dryer are negligible. Properties The gas constant of air is R 0.87 kpa.m /kg.k (Table A-1. Also, C p kj/kg K for air at room temperature (Table A-15. Analysis (a We take the hair dryer as the system. This is a control volume since mass crosses the system boundary during the process. We observe that this is a steady-flow process since there is no change with time at any point and thus ΔmCV 0 and ΔECV 0, and there is only one inlet and one exit and thus m& m& m&. The energy balance for this steady-flow system can be expressed in the rate form as 1 & & & 0 (steady E E ΔE E& E& 1 system 0 1 in out in out Rate of net energy transfer by heat, work, and mass Rate of change in internal, kinetic, potential, etc. energies & & 0 W + W + mh & 0 + mh & (since Δke Δpe 0 e,in fan,in Thus, W& m& C p Then, e,in 1 out W& mc & ( T T e,in p 1 ( T T ( kj/kg C( 7 v 1 V& 1 1 RT P 1 1 mv & 1 1.kJ/s ( 0.87kPa m /kg K( 95K ( 100kPa kg/s C 0.867m ( kg/s( 0.867m /kg 0.00m /s (b The exit velocity of air is determined from the conservation of mass equation, 1 m& v v RT P mv & ( kg/s( m /kg AV V 7.0 m/s A m /kg T 7 C A 60 cm (0.87 kpa m /kg K(0 K m /kg (100 kpa W e 100 W P kpa T 1 C 1-19

22 1-1 The ducts of an air heating system pass through an unheated area, resulting in a temperature drop of the air in the duct. The rate of heat loss from the air to the cold environment is to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of -11 C and.77 MPa. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air. This assumption results in negligible error in heating and air-conditioning applications. Properties The specific heat of air at room temperature is C p kj/kg C (Table A-15. Analysis We take the heating duct as the system. This is a control volume since mass crosses the system boundary during the process. We observe that this is a steady-flow process since there is no change with time at any point and thus ΔmCV 0 and ΔECV 0. Also, there is only one inlet and one exit and thus m& m& m&. The energy balance for this steady-flow system can be expressed in the rate form as 1 & & & 0 (steady E E ΔE E& E& 1 system 0 1 in out in out Rate of net energy transfer by heat, work, and mass Substituting, & Rate of change in internal, kinetic, potential, etc. energies mh & & 1 Qout + mh & (since Δke Δpe 0 mc & ( T T out p 1 ( 10kg/min( 1.007kJ/kg C( C 6 kj/min out mc p ΔT 10 kg/min AIR Q 1-0

23 1-E Air gains heat as it flows through the duct of an air-conditioning system. The velocity of the air at the duct inlet and the temperature of the air at the exit are to be determined. Assumptions 1 Air is an ideal gas since it is at a high temperature and low pressure relative to its critical point values of - F and 58 psia. The kinetic and potential energy changes are negligible, Δke Δpe 0. Constant specific heats at room temperature can be used for air, C p 0.0 and C v Btu/lbm R. This assumption results in negligible error in heating and air-conditioning applications. Properties The gas constant of air is R 0.70 psia.ft /lbm.r (Table A-1. Also, C p 0.0 Btu/lbm R for air at room temperature (Table A-15E. Analysis We take the air-conditioning duct as the system. This is a control volume since mass crosses the system boundary during the process. We observe that this is a steady-flow process since there is no change with time at any point and thus ΔmCV 0 and ΔECV 0, there is only one inlet and one exit and thus m& 1 m& m&, and heat is lost from the system. The energy balance for this steady-flow system can be expressed in the rate form as & & & 0 (steady E E ΔE E& E& 1 system 0 1 in out in out Rate of net energy transfer by heat, work, and mass Rate of change in internal, kinetic, potential, etc. energies in + mh & 1 mh & (since Δke Δpe 0 mc & ( T T in p 1 (a The inlet velocity of air through the duct is determined from V & 1 V& 1 50 ft /min V1 85 ft/min A πr π 5/1 ft 1 ( (b The mass flow rate of air becomes 1 ( 0.70psia ft /lbm R( 510R ( 15psia RT1 v1 1.6 ft P1 V& 1 50ft / min m& 5.7lbm/min lbm/ s v 1.6ft /lbm / lbm Then the exit temperature of air is determined to be in Btu/s T T F F mc & 0.595lbm/s 0.0Btu/lbm F p ( ( 50 ft /min AIR Btu/s D 10 in 1-1

24 1- Water is heated in an insulated tube by an electric resistance heater. The mass flow rate of water through the heater is to be determined. Assumptions 1 Water is an incompressible substance with a constant specific heat. The kinetic and potential energy changes are negligible, Δke Δpe 0. Heat loss from the insulated tube is negligible. Properties The specific heat of water at room temperature is C p.18 kj/kg C (Table A-9. Analysis We take the tube as the system. This is a control volume since mass crosses the system boundary during the process. We observe that this is a steady-flow process since there is no change with time at any point and thus ΔmCV 0 and ΔECV 0, there is only one inlet and one exit and thus m& 1 m& m&, and the tube is insulated. The energy balance for this steady-flow system can be expressed in the rate form as Thus, & & & 0 (steady E E ΔE E& E& 1 system 0 1 in out in out Rate of net energy transfer by heat, work, and mass Rate of change in internal, kinetic, potential, etc. energies W& + mh & mh & (since Δke Δpe 0 e,in W& m& C T W& mc & ( T T e,in ( T e,in 1 1 p 1 ( 7 kj/s (.18 kj/kg C( kg/s C WATER 15 C 70 C 7 kw 1-

25 Heat Transfer Mechanisms 1-C The thermal conductivity of a material is the rate of heat transfer through a unit thickness of the material per unit area and per unit temperature difference. The thermal conductivity of a material is a measure of how fast heat will be conducted in that material. 1-5C The mechanisms of heat transfer are conduction, convection and radiation. Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas which is in motion, and it involves combined effects of conduction and fluid motion. Radiation is energy emitted by matter in the form of electromagnetic waves (or photons as a result of the changes in the electronic configurations of the atoms or molecules. 1-6C In solids, conduction is due to the combination of the vibrations of the molecules in a lattice and the energy transport by free electrons. In gases and liquids, it is due to the collisions of the molecules during their random motion. 1-7C The parameters that effect the rate of heat conduction through a windowless wall are the geometry and surface area of wall, its thickness, the material of the wall, and the temperature difference across the wall. 1-8C Conduction is expressed by Fourier's law of conduction as Q & ka dt cond where dx dt/dx is the temperature gradient, k is the thermal conductivity, and A is the area which is normal to the direction of heat transfer. Convection is expressed by Newton's law of cooling as Q & conv has ( Ts T where h is the convection heat transfer coefficient, A s is the surface area through which convection heat transfer takes place, T s is the surface temperature and T is the temperature of the fluid sufficiently far from the surface. Radiation is expressed by Stefan-Boltzman law as Q & rad εσas ( Ts Tsurr where ε is the emissivity of surface, A s is the surface area, T s is the surface temperature, T surr is average surrounding surface temperature and σ W/m.K is the Stefan- Boltzman constant. 1-9C Convection involves fluid motion, conduction does not. In a solid we can have only conduction. 1-50C No. It is purely by radiation. 1-

26 1-51C In forced convection the fluid is forced to move by external means such as a fan, pump, or the wind. The fluid motion in natural convection is due to buoyancy effects only. 1-5C Emissivity is the ratio of the radiation emitted by a surface to the radiation emitted by a blackbody at the same temperature. Absorptivity is the fraction of radiation incident on a surface that is absorbed by the surface. The Kirchhoff's law of radiation states that the emissivity and the absorptivity of a surface are equal at the same temperature and wavelength. 1-5C A blackbody is an idealized body which emits the maximum amount of radiation at a given temperature and which absorbs all the radiation incident on it. Real bodies emit and absorb less radiation than a blackbody at the same temperature. 1-5C No. Such a definition will imply that doubling the thickness will double the heat transfer rate. The equivalent but more correct unit of thermal conductivity is W.m/m. C that indicates product of heat transfer rate and thickness per unit surface area per unit temperature difference. 1-55C In a typical house, heat loss through the wall with glass window will be larger since the glass is much thinner than a wall, and its thermal conductivity is higher than the average conductivity of a wall. 1-56C Diamond is a better heat conductor. 1-57C The rate of heat transfer through both walls can be expressed as T1 T T1 T wood k wood A (0.16 W/m. C A 1.6A( T1 T Lwood 0.1m T1 T T1 T brick k brick A (0.7 W/m. C A.88A( T1 T Lbrick 0.5 m where thermal conductivities are obtained from table A-5. Therefore, heat transfer through the brick wall will be larger despite its higher thickness. 1-58C The thermal conductivity of gases is proportional to the square root of absolute temperature. The thermal conductivity of most liquids, however, decreases with increasing temperature, with water being a notable exception. 1-59C Superinsulations are obtained by using layers of highly reflective sheets separated by glass fibers in an evacuated space. Radiation heat transfer between two surfaces is inversely proportional to the number of sheets used and thus heat loss by radiation will be very low by using this highly reflective sheets. At the same time, evacuating the space between the layers forms a vacuum under atm pressure which minimize conduction or convection through the air space between the layers. 1-

27 1-60C Most ordinary insulations are obtained by mixing fibers, powders, or flakes of insulating materials with air. Heat transfer through such insulations is by conduction through the solid material, and conduction or convection through the air space as well as radiation. Such systems are characterized by apparent thermal conductivity instead of the ordinary thermal conductivity in order to incorporate these convection and radiation effects. 1-61C The thermal conductivity of an alloy of two metals will most likely be less than the thermal conductivities of both metals. 1-6 The inner and outer surfaces of a brick wall are maintained at specified temperatures. The rate of heat transfer through the wall is to be determined. Assumptions 1 Steady operating conditions exist since the surface temperatures of the wall remain constant at the specified values. Thermal properties of the wall are constant. Properties The thermal conductivity of the wall is given to be k 0.69 W/m C. Brick Analysis Under steady conditions, the rate of heat transfer through the wall wall is ΔT (0 5 C cond ka (0.69W/m C(5 6m 105W L 0.m 0. m 0 cm 0 C 5 C 1-6 The inner and outer surfaces of a window glass are maintained at specified temperatures. The amount of heat transfer through the glass in 5 h is to be determined. Assumptions 1 Steady operating conditions exist since the surface temperatures of the glass remain constant at the specified values. Thermal properties of the glass are constant. Properties The thermal conductivity of the glass is given to be k 0.78 W/m C. Analysis Under steady conditions, the rate of heat transfer through the glass by conduction is ΔT (10 C cond ka (0.78 W/m C( m 68 W L 0.005m Glas Then the amount of heat transfer over a period of 5 h becomes Q cond Δt (.68 kj/s(5 600 s 78,60 kj If the thickness of the glass doubled to 1 cm, then the amount of heat transfer will go down by half to 9,10 kj. 10 C

28 1-6 "GIVEN" "L0.005 [m], parameter to be varied" A* "[m^]" T_110 "[C]" T_ "[C]" k0.78 "[W/m-C]" time5*600 "[s]" "ANALYSIS" Q_dot_condk*A*(T_1-T_/L Q_condQ_dot_cond*time*Convert(J, kj L [m] Q cond [kj]

29 Q cond [kj] L [m] 1-7

30 1-65 Heat is transferred steadily to boiling water in the pan through its bottom. The inner surface of the bottom of the pan is given. The temperature of the outer surface is to be determined. Assumptions 1 Steady operating conditions exist since the surface temperatures of the pan remain constant at the specified values. Thermal properties of the aluminum pan are constant. Properties The thermal conductivity of the aluminum is given to be k 7 W/m C. Analysis The heat transfer area is A π r² π(0.1 m² 0.01 m² Under steady conditions, the rate of heat transfer through the bottom of the pan by conduction is Δ &Q ka T ka T T 1 L L Substituting, 105 T 105 C 800W (7W/m C(0.01m 0.00m which gives T 105. C 1-66E The inner and outer surface temperatures of the wall of an electrically heated home during a winter night are measured. The rate of heat loss through the wall that night and its cost are to be determined. Assumptions 1 Steady operating conditions exist since the surface temperatures of the wall remain constant at the specified values during the entire night. Thermal properties of the wall are constant. Properties The thermal conductivity of the brick wall is given to be k 0. Btu/h.ft. F. Analysis (a Noting that the heat transfer through the wall is by conduction and the surface area of the wall is A 0 ft 10 ft 00 ft, the steady rate of heat transfer through the wall can be determined from ka T 1 T (. ( 6 5 F 0 Btu / h. ft. F(00 ft 108 Btu / h L 1 ft or kw since 1 kw 1 Btu/h. (b The amount of heat lost during an 8 hour period and its cost are Q Q & Δt ( kw(8 h 7.88 kwh Cost (Amount of energy(unit cost of energy (7.88 kwh($0.07 / kwh $0.51 Brick Q Therefore, the cost of the heat loss through the wall to the home owner that night is $ ft 6 F 5 F 1-8

31 1-67 The thermal conductivity of a material is to be determined by ensuring onedimensional heat conduction, and by measuring temperatures when steady operating conditions are reached. Assumptions 1 Steady operating conditions exist since the temperature readings do not change with time. Heat losses through the lateral surfaces of the apparatus are negligible since those surfaces are well-insulated, and thus the entire heat generated by the heater is conducted through the samples. The apparatus possesses thermal symmetry. Analysis The electrical power consumed by the heater and converted to heat is W & e VI ( 110 V ( 0. 6 A 66 W The rate of heat flow through each sample is & W e 66 W W Then the thermal conductivity of the sample becomes πd π(. 00 m A m Δ ka T QL & ( W(0.0 m k 78.8 W / m. C L AΔT ( m ( 10 C Q cm cm 1-68 The thermal conductivity of a material is to be determined by ensuring onedimensional heat conduction, and by measuring temperatures when steady operating conditions are reached. Assumptions 1 Steady operating conditions exist since the temperature readings do not change with time. Heat losses through the lateral surfaces of the apparatus are negligible since those surfaces are well-insulated, and thus the entire heat generated by the heater is conducted through the samples. The apparatus possesses thermal symmetry. Q & Analysis For each sample we have Q & 5 / W A (. 01m( 01. m 001. m ΔT C Then the thermal conductivity of the material becomes & & Δ (. Q ka T QL 17 5 W(0.005 m k 1.09 W / m. C L AΔT (. 001m ( 8 C A L L 1-9